Portfolio Efficient Set
Niraj Sardar
Loading libraries
library(quantmod)## Warning: package 'quantmod' was built under R version 4.1.3## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric## Loading required package: TTR## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(ggplot2)## Warning in register(): Can't find generic `scale_type` in package ggplot2 to
## register S3 method.library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:xts':
##
## first, last## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionticks <- c("STZ", "CAT", "LLY", "CBRL")
retout <- NULL
retout <- xts(retout)
for(i in 1:length(ticks)){
prices = getSymbols(ticks[i], auto.assign = F)
returns <- periodReturn(prices, period = "monthly",
type = "arithmetic")
retout <- merge.xts(retout, returns)
}
colnames(retout) <- ticks
retout <- retout['2010/2015']
retout <- na.omit(retout)
head(retout)## STZ CAT LLY CBRL
## 2010-01-28 19:00:00 0.009416196 -0.08334795 -0.01428166 -0.02711248
## 2010-02-25 19:00:00 -0.064676617 0.09207498 -0.02443185 0.18181821
## 2010-03-30 20:00:00 0.093085173 0.10166519 0.05474668 0.06181321
## 2010-04-29 20:00:00 0.111313801 0.08337308 -0.03451132 0.06446740
## 2010-05-27 20:00:00 -0.088122605 -0.10765161 -0.06233915 0.00931746
## 2010-06-29 20:00:00 -0.062424970 -0.01135612 0.02165291 -0.06562314B. Calculate means and covariance matrix
meanret <- colMeans(retout,na.rm = T)
x1 = round(meanret, 5)
cat("The mean vector :\n")## The mean vector :print(x1)## STZ CAT LLY CBRL
## 0.03421 0.00581 0.01275 0.01885covar <- var(retout)
x2 = round(covar, 8)
cat("The covariance matrix: \n")## The covariance matrix:print(x2)## STZ CAT LLY CBRL
## STZ 0.00745920 0.00105944 0.00058250 0.00119845
## CAT 0.00105944 0.00699707 0.00031562 0.00100841
## LLY 0.00058250 0.00031562 0.00154920 0.00029047
## CBRL 0.00119845 0.00100841 0.00029047 0.00405279C. Creating one portfolio as an example to showcase method
The weight vector contains weights for each security. Weights add to 1.
weight <- c(.2, .3, .3, .2)Calculate portfolio variance Using matrix algebra
weight <- as.matrix(weight)
dim(weight)## [1] 4 1meanret <- as.matrix(meanret)
dim(meanret)## [1] 4 1mretp <- t(weight) %*% meanret
sretp <- sqrt(t(weight) %*% covar %*% weight)
cat("The mean and sigma of portfolio returns: ", mretp, sretp)## The mean and sigma of portfolio returns: 0.01617978 0.0416561D. Run simulation with random weights
First, for niter iterations, let’s create random portfolio weights.
set.seed(12)
niter <- 500 # Set the number of iterations here
randomnums <- data.frame(replicate(4, runif(niter, 1, 10)))
head(randomnums)## X1 X2 X3 X4
## 1 1.624248 3.921433 1.545892 6.809350
## 2 8.359977 4.151757 2.406675 8.817655
## 3 9.483596 5.183035 8.274220 3.444403
## 4 3.424437 2.660027 5.040517 1.732409
## 5 2.524133 5.642371 5.675101 5.134248
## 6 1.305061 7.567241 6.264320 3.112291wt_sim <- randomnums / rowSums(randomnums)
cat("The weights after normalization are in wt_sim...")## The weights after normalization are in wt_sim...head(wt_sim)## X1 X2 X3 X4
## 1 0.11684463 0.2820988 0.1112079 0.4898488
## 2 0.35220570 0.1749135 0.1013932 0.3714877
## 3 0.35942787 0.1964368 0.3135926 0.1305427
## 4 0.26633999 0.2068870 0.3920327 0.1347403
## 5 0.13301817 0.2973448 0.2990696 0.2705674
## 6 0.07151443 0.4146680 0.3432709 0.1705466Initializing Variables.
# initialize weight and Results matrices
weight <- matrix(data = NA, nrow = length(ticks), ncol = 1)
Results <- matrix(data = NA, nrow = niter, ncol = 6)Run the simulations - this means, do portfolio calculations for each simulated portfolio.
# loop: each i is a portfolio
for (i in 1:niter){
# inner loop places weights into Results
for (k in 1:length(ticks)) {
Results[i,k] = weight[k,1] = wt_sim[i,k]
}
Results[i,5] <- t(weight) %*% meanret # portfolio mean
Results[i,6] <- sqrt(t(weight) %*% covar %*% weight) # portfolio sigma
}
colnames(Results) <- c(ticks, "PortMean", "PortSigma")
Results <- as.data.frame(Results)
head(Results)## STZ CAT LLY CBRL PortMean PortSigma
## 1 0.11684463 0.2820988 0.1112079 0.4898488 0.01628776 0.04693199
## 2 0.35220570 0.1749135 0.1013932 0.3714877 0.02136079 0.04862590
## 3 0.35942787 0.1964368 0.3135926 0.1305427 0.01989614 0.04430372
## 4 0.26633999 0.2068870 0.3920327 0.1347403 0.01785103 0.04003521
## 5 0.13301817 0.2973448 0.2990696 0.2705674 0.01519099 0.04083742
## 6 0.07151443 0.4146680 0.3432709 0.1705466 0.01244703 0.04391907Plotting the results.
ggplot(data = Results , aes(x = PortSigma, y = PortMean)) +
geom_point(pch = 10, colour = "red", size = 3)
E. Optimization
Again, risk is bad while return is good, so one way to select the best portfolio(s) is to identify a constraint based on one of these dimensions, and locate the best portfolio using the other.
minmret = min(Results$PortMean)
maxmret = max(Results$PortMean)
seqmret = seq(round(minmret,3)-.001, maxmret+.001, .001)
optim <- Results %>% mutate(portnumber = index(Results)) %>%
mutate(ints = cut(PortMean ,breaks = seqmret),
lower = as.numeric( sub("\\((.+),.*", "\\1", ints) )) %>%
group_by(ints) %>%
summarise( lowerval = min(lower),
sig_optim = min(PortSigma),
retn_optim = PortMean[which.min(PortSigma)],
numb = length(PortSigma),
portID=portnumber[which.min(PortSigma)])
optim## # A tibble: 17 x 6
## ints lowerval sig_optim retn_optim numb portID
## <fct> <dbl> <dbl> <dbl> <int> <int>
## 1 (0.011,0.012] 0.011 0.0449 0.0119 1 446
## 2 (0.012,0.013] 0.012 0.0405 0.0129 11 407
## 3 (0.013,0.014] 0.013 0.0364 0.0137 20 315
## 4 (0.014,0.015] 0.014 0.0347 0.0148 45 231
## 5 (0.015,0.016] 0.015 0.0353 0.0151 57 242
## 6 (0.016,0.017] 0.016 0.0376 0.0162 48 352
## 7 (0.017,0.018] 0.017 0.0358 0.0171 76 282
## 8 (0.018,0.019] 0.018 0.0370 0.0180 72 345
## 9 (0.019,0.02] 0.019 0.0382 0.0191 68 403
## 10 (0.02,0.021] 0.02 0.0404 0.0201 42 218
## 11 (0.021,0.022] 0.021 0.0426 0.0211 28 485
## 12 (0.022,0.023] 0.022 0.0466 0.0220 14 76
## 13 (0.023,0.024] 0.023 0.0500 0.0236 10 139
## 14 (0.024,0.025] 0.024 0.0528 0.0247 4 495
## 15 (0.025,0.026] 0.025 0.0563 0.0255 2 36
## 16 (0.026,0.027] 0.026 0.0631 0.0269 1 198
## 17 (0.027,0.028] 0.027 0.0627 0.0271 1 120ggplot(data = optim , aes(x = sig_optim, y = retn_optim)) +
geom_point(pch = 10, colour = "red", size = 3)